THE SIDES AND ANGLES OF A TRIANGLE

Book I. Propositions 18 and 19

THE STUDENT by now must have some appreciation of what is involved in a logical theory. Each proposition must wait its turn, for each depends on some previous proposition or first principle -- even the most obvious.

Consider for example the following:

Any two sides of a triangle are togethergreater than the remaining side.

In other words, a straight line is the shortest distance between two points!

If anyone wanted to ridicule mathematics for its insistence on the axiomatic method of orderly proof, this theorem offers a wide target. In fact, the Epicureans (those Athenian free-thinkers, who defined philosophy as the art of making life happy) did exactly that. They said that this theorem required no proof, and was known even to an ass. For if hay were placed at one vertex, they argued, and an ass at another, the poor dumb animal would not travel two sides of the triangle to get his food, but only the one side which separated them.

Such is the scorn that the true philosopher must bear! And what
can the mathematician do but to point out, patiently, that mathematics as a logical science relies on deduction from first principles. Those principles moreover should be as few in number as possible — whatever can be proved should be. That is the intellectual sport.

This proposition -- Any two sides of a triangle are together greater than the remaining side -- is Euclid's Proposition 20. And to prove it he first had to prove

Proposition 19. A greater angle of a triangle is opposite a greater side.

If angle A is greater than angle B, then side CB is greater than side CA.

And to prove that he first had to prove

Proposition 18. A greater side of a triangle is opposite a greater angle.